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NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
TECHNICAL MEMORANDUM 1314
ON THE TURBULENT FRICTION LAYER FOR RISING PRESSURE
By K. Wieghardt and W. TiUmaim
Translation of ZWB Untersuchungen und Mitteilungen
Nr. 6617, November 20, 1944
NACA
Washington
October 1951
..NTSDEPAR-mEMT
;,ARSTON SCIENCE UBRARY
RO. BOX 117011 QiiuS>.
GAINESVILLE. FL 3^t>i
^VS
iC i^^HO ^iiir'ir
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL MEMORANDUM 131^
ON THE TURBULENT FRICTION LAYER FOR RISING PRESSURE*
By K. Wieghardt and W. TilLmann
Abstract: As a supplement to the UM report 6603, measurements in tur-
bulent friction layers along a flat plate with 'rising pres-
sure are further evaluated. The investigation was performed
on behalf of the Aerodynamischen Versuchsanstalt Gottingen.
Outline: 1. SYMBOLS
2. INTRODUCTION
3. TEST SETUP
k. TEST RESULTS
5. ON THE GRUSCHWITZ CALCULATION METHOD
6. ON AN ENERGY THEOREM FOR FRICTION LAYERS
7. SUMMARY
8. REFERENCES
1. SYMBOLS
X position rearward from leading edge of the plate
y distance from wall
u velocity component in x-direction
V velocity component in y-direction
U^V velocity components outside of the friction layer
p density
\i viscosity \, of the air-
V kinematic viscosity
p static pressure
"Zur tvirbulenten Reibungsschicht bei Druckanstieg." Zentrale fiir
wissenschaftliches Berichtswesen der Luftfahrtforschung des Generalluft-
zeugmeisters (ZWB), Berlin-Adlershof, Untersuchungen und Mitteilungen
Nr. 6617, Kaiser Wilhelm-Institut fiir Stromungsforschung, Gottingen,
November 20, 19iiU.
NACA TM ISl^^
P 2
q = 2U
= p + q
dynamic pressure
total pressure
D ?
Q = ^ U dynamic pressure outside of the friction layer
Ux/v
U62/V
&
Reynolds number
Reynolds number of the friction layer
friction layer thickness
6-, = / (1 - 77)^ displacement thickness
^0
^6
62 =
^3 =
u
momentum loss thickness
dy energy loss thickness
%2 - ^l/S^
Hop — Oo/Oq
^ form parameters of the velocity profile
T) = 1 -
"u(S2)"
U
^•f'
= T
turbulent shearing stress
wall shearing stress
/p p
j/p U"^ local friction drag coefficient
mixing length
2. INTRODUCTION
For the calculation of laminar boundary layers in two-dimensional
incompressible flow, numerous calculation methods have been developed
NACA TM 131^
on the basis of Prandtl's boundary -layer equation; in contrast, only the
semiempirical method of E. Gruschwitz (reference l) and its improvements
by A. Kehl (reference 2) and A. Walz (references 3^ ^, and 5) are avail-
able for turbulent friction layers. The reason, as is well known, lies
in the lack of a mathematical law for the apparent shearing stress t
which originates by the turbulent mixing of momentum. Prandtl's expres-
sion for the mixing length so far has led to success only in cases where
a sufficient niJinber of correct (and also sufficiently simple) data on the
variation of the mixing length can be obtained directly from the geometry
of the flow as for instance in the case of the free jet. 'As a basis for
the Gruschwitz method there serve, therefore, besides the momentum equa-
tion for friction layers, three statements which are partly empirical
and lack a theoretical basis. They are as follows:
A. The velocity profiles in the turbulent friction layer for vari-
able outside pressure form, after having been made adequately dimension-
less, a single-parameter family of ciirves, if one disregards the laminar
sublayer; thus, every profile may be characterized by a single
quantity (t^) .
B. A differential equation, likewise derived solely from experi-
ments, concerning the variation of this parameter in flow direction as
a function of the pressure variation and of the momentum thickness.
C. An assumption concerning the wall shearing stress. Gruschwitz
inserts a constant as first approximation.
A. Kehl (reference 2) then improved the Gruschwitz method. According
to his measurements which extended over a larger Re-number range than
those of Gruschwitz, it is necessary to insert in statement B, for a
higher Re-number of the friction layer, a function of the Re-number
U52/V instead of a certain constant b. Furthermore, Kehl obtained
better agreement between the calculation and his test results by sub-
stituting in scatement C for the wall shearing stress the value which
results for the respective Re(52) number at a flat plate with constant
outside pressure. Finally, A. Walz (references 3^ ^) and 5) greatly
simplified the integration method mathematically so that for prescribed
variation of the velocity outside of the friction layer U(x) the mom-
entum thickness, the form parameter t\, and hence the point of separ-
ation can be calculated very quickly. No statement is obtained regarding
the wall shearing stress, since it had, on the contrary, been necessary
to make the assumption C concerning Tq in order to set up the cal-
culation method at all.
Thus it seemed desirable to investigate the friction drag of a
smooth plate for variable outside pressure. On one hand, direct interest
in this exists in view of the wing drag; on the other hand, one could
NACA TM 131^
expect an Improvement in the above calculation method if an accurate
statement regarding Tq could be substituted for the assumption C. In
order to arrive at the simplest possible laws, friction layers along a
flat smooth plate were investigated where a systematically increasing
pressure was produced by an opposing plate. By analogy with the behavior
of laminar boundary layers the wall shearing stress was expected to
decrease in the flow direction up until separation, more strongly than
In case of constant outside pressure. Instead, Tq increased, after a
certain starting distance, more or less suddenly to a multiple of the
initial value. A brief report on this striking behavior of the friction
drag has already been published (reference 6) . In the present report,
these tests are further evaluated and compared with those of Gruschwitz
and Kehl. Considerable deviations result in places; however, it was not
possible to develop a better calculation method with this new test mate-
rial either.
Since the application of an energy theorem had proved expedient for
the calc\ilation of laminar boundary layers (reference 'j) , a theoretical
attempt in this respect was made for turbulent friction layers, too;
however, it did not meet with the same success. Merely an interpretation
for the statement B can be obtained in this manner, which is. however,
not cogent.
3. TEST SETUP
The test setup and program have been described in the preliminary
report. A new measuring method was developed where with the aid of a
pressure rake and of a multiple manometer (reference 8) turbulent friction
layers co\ild be measured quickly and accurately and the computational
evaluation greatly simplified.
Friction layers were measured at p = for two different veloc-
ities U = const., four cases, I to IV, with rising, and one, V, with
diminishing pressure (figs. 1 to 6) . In cases I and II, p increases
almost in the entire measuring range linearly with rearward position
with respect to the leading edge of the plate x, whereas in cases III
and rv the outer velocity U (U ~ x"^) decreases with a power of x
(figs. 1 to 6) .
The velocities were about 20 to 60 meters per second; the test
section length was 5 meters so that Re -numbers up to 107 were attained.
The Re-number of the friction layer (formed with the momentum thickness)
increased to from 2 to 7 X lo5.
MCA TM IBlij-
h. TEST RESTTLTS
Mainly the variation of the wall shearing stress had heen described
in the brief preliminary report (reference 6); later the measurements
were evaluated more thoroughly. First, we plotted figures 1 to 6 for
the different pressure variations: the outer velocity U, the local
friction drag coefficient c^', the displacemenx thickness 6^, the
momentum thickness 82^ and the energy loss thickness 5^ .treated in
section 6; in analogy to the momentum loss thickness 82^ 80 is a
measure of how much kinetic energy of the flow is lost mechanically due
to the friction layer, that is, is converted to heat by the effect of
the friction forces.
In case of constant outside pressure Cf' depends, for a smooth
plate, only on the Re-number. The test points from figure 1 lie between
the formulas for cf' according to L. Prandtl (reference 9)^ F. Schultz-
Grunow (reference 10 ), and J. Nikuradse (reference 11) which in the test
range Ux/v = 3 X lO-' to 10' differ only slightly. For rising pressure,
a slight decrease of c^' with rearward position results at first,
according to figures 2 to 5j however, after a certain distance c^ '
increases more or less suddenly to a multiple of its original amount and
decreases again only at the end of the test section where for test tech-
nical reasons the pressure increase could no longer be maintained.
W. Mangier (reference 12) found the same unexpected behavior in further
evaluating the measurements of A. Kehl (reference 2). In contrast, c^'
varies only slightly in case of decreasing pressure. Thus, on one hand,
-T*- > must be responsible for the strong increase in wall shearing
stress; furthermore, the Re-number of the friction layer is of importance
since first a certain starting distance is required. Therefore, we
plotted in figure 7 c^' against the dimensionless -r- — ^. Our measure-
■'■ y dx
ments resulted in a comparatively narrow bundle of curves; however, the
results according to Kehl -Mangier, drawn in in dashed lines, cannot be
brought under a common denominator in this manner. The test arrangement
of Kehl was more general insofar as he had at first a piece of laminar
boundary layer and, moreover, in some cases first a pressure drop, and
then an adjoining pressure rise; in our tests, in contrast, the friction
layer, starting from the leading edge of the plate, had been made tur-
bulent by a trip wire and the pressure increased monotonically.
NACA TM 13114-
In want of a better criterion, we can see from figure 7 that the
strong increase in friction drag is not to be expected as long as
^ ^ < 2 X 10-3
Q dx
The displacement and momentum thickness as well as the wall shearing
stress represent only a summary of the development of a friction layer.
Therefore, we shall consider below the velocity profiles and shearing
stress profiles. The variation of — against the rearward position
from the leading edge for constant wall distance is particularly illus-
trative (figs. 8 and 9)- In case II (fig. 8), — suddenly drops steeply
in the layers near the wall whereas it decreases continuously in case IV.
Accordingly, the characteristic lengths S]_, 62, and 60 increase at
these points more strongly than before. Since the drag coefficient Cf'
depends essentially on the variation of ^— , one recognizes at once in
ox
case II the point of maximum wall shearing stress at x w 3'3 meters.
The shearing stresses are obtained from Prandtl's boundary- layer
equation which may be transformed with the aid of the continuity equation
and of the Bernoulli equation = -UU„ valid outside of the friction
p dx ^
layer. With U^ = -p , % = v^, Uy = ^, and Q = ^ U , one obtains
S_i_^,li^x_!^ r^^ay (1)
Sy 2Q U U U U Jq U
The integration which is still to be performed yields additionally a
control value for the wall shearing stress
— = = - / dy (2)
2Q 2 Jq ^y 2Q
However, the value for Tq obtained in this manner is not as reliable
as the one calculated from the momentum theorem because here graphical
differentiation is applied more often. According to a suggestion by
MCA TM 131^
Professor Betz, the momentum theorem may be transformed for the calcvil-
atlon of Tq in the following manner (compare reference 6):
2Q y2+Hi2 dx\
-.,) . (h,, - H,,)i 3^ a, (3)
where a suitable mean value of E-^Q ^ ^l/^2 ^^ substituted for H2_2-
Then the second term is small compared to the first and plays only the
role of a correction term so that essentially only one graphical differ-
entiation has to be performed.
First, the shearing stress profiles are plotted for constant outside
pressure in figure 10. The shearing stress t with the appertaining
wall shearing stress t^ and the wall distance y with the momentum
thickness S2 are made diraensionless. These dimensionless profiles are
almost completely identical although a systematic variation with the
Re-number is recognizable. The profiles t against y for the two
cases II and IV with press\ire rise follow in figures 11 and 13;
figures 12 and 1^4- show the corresponding dimensionless shearing stress
profiles t/tq against y/52- In the case II where the pressure p
increases approximately linearly, the t -profiles differ considerably
for various rearward positions, especially the wall distance at which
T reaches its maximum value is subject to great changes. In contrast,
-0 2T
the dimensionless profiles in case IV where U <*< 53(x/l°') ' meters
per second is valid are very similar. Only the one profile at
X = 1.99 meters stands out sharply without any perceptible reason.
According to D. R. Hartree (reference 13) ^ similar velocity and hence
also shearing stress profiles result for laminar boundary layer in the
case U ~ x~ since for laminar flow t is simply t = U ^. In the
turbulent friction layer, the velocity and hence also the shearing stress
profiles thus vary in this case; however, the latter at least can be
described - although somewhat forcibly - by a single -parameter family of
curves, according to figure ik.
A. Buri (reference 1^) attempted to interpret the shearing stress
profiles in general as single -parameter family of curves. Bvurl selected
for the parameter Tg^ the computed wall tangent of the dimensionless
S2/St'
shearing stress profile ?„ = — \^—] because for this quantity a
^o\ay/y^o
relation may be read off immediately from the boiindary-layer equation.
8 NACA TM 131^
For y=0, u=v=0 and therewith r„ = ^ which yields
^ To dx -^
-— I = -^. The tangent to the shearing stress profile thus calcu-
^y/y=o ^"^
lated is known in general to fit very hadly the shearing stress variation
in the proximity of the wall determined from measurements. In the tur-
bulent friction layer, the velocity u decreases only in immediate wall
proximity - in the so-called lajninar sublayer - to zero so that the tan-
gent direction deviates from the above calculated value even for very
small wall distances. This is shown also in figure 15, which represents
the shearing stress profiles in wall proximity and the pertaining wall
tangents found by calculation; for constant outside pressure, too, the
curves against y apparently begin to drop from the point y = out-
ward with a definite angle rather than continue with a horizontal
tangent (fig. 10). Nevertheless, Fg^ could be used at first as a com-
puted quantity for the characterization of a definite shearing stress
profile. However, in case II, for instance, the same shearing stress
profile would have to be present for x = 3-19 meters and for
X = 3-79 meters according to figure 15, which is obviously not true
according to figure 12. In general, the shearing stress profiles could
not be characterized by one other parameter alone either. At least two
quantities would be required for this, for instance, the magnitude and
the dimensionless wall distance of the maximum ^jj^^^/tq.
Finally, we calculated from the shearing stress the mixing lengths
5u
according to Pranatl's expression t = p2 -—
oy
and plotted them in
Sy
figures I6 to 19 against the wall distance y or in dimensionless form
Z/62 against y/52. Like the thickness of the friction layer, 2
increases more and more with further rearward movement; in case IV, 2
even increases on approaching the end of the test section to 30 milli-
meters. The diminishing of the mixing length for large wall distances
cannot be specified, as is well known, since 2 there is computed as
the quotient of two small quantities, namely, t and r— • Here
again I increases linearly in wall proximity. In the tube, the result
had been 2 = O.^J-y and at the plate for constant pressure, according
to Schult^ -Grunow (reference 10) 2 = 0.i4-3y (dashed in figs. I7 and 19)-
Here, at rising pressure, I increases at first again linearly but far
more rapidly. In case II, 2 attains the maximum value 2 = l.ly and
in case IV even 2 = 2.0y. Although no fixed relation exists between
2/y in wall proximity and the wall shearing stress, it is striking in
figures I7 and 19 that 2/y is largest just for those rearward positions
where the wall shearing stress Tq also (compare figs. 3 and 5) attains
its maximum value.
NACA TM 131^
"i. ON THE GRUSCHWITZ CALCULATION METHOD
Our test material enables us to examine the basis of the Gruschwitz
method enumerated under section 1. The assumption A according to which
all turbulent velocity profiles concerned form a single -parameter family
of curves is again confirmed in figure 20. Here a few velocity profiles
of the different cases I to V are plotted for which the evaluation
accidentally had resulted in exactly the same ratio H2^2 ~ ^1/^2^' ^^^
profiles u/U against y/Sg with equal H12 ^^^ hence equal t\
(compare fig. 27) are in agreement even for different velocities^ rear-
ward positions^ and pressure variations, thus also after different
"previous history." This single -parameter quality covers, however, only
the "visible" turbulent part of the profile; the velocities in the
laminar sublayer may have a different distribution even for equal t];
otherwise, a unique connection would necessarily exist between the wall
tangent Fg^ and i^ or Hj_2 which is certainly not the case according
to what was said above .
The change of the velocity profiles characterized by the param-
eters r| or H2^2 with change in the rearward position is represented
in figures 21 and 22 for all measuring series. The weak but throughout
systematic dependence of the profile on the Re -number for constant out-
side pressure is remarkable; Niku_adse (reference 11), on the other
hand, obtained here always the same profile with t] = 0.515 and
H;L2 = 1-302.
In order to apply the momentum theorem for the Gruschwitz method,
the relation between H]_2 and t), necessarily unique for a single-
parameter profile class, must be known. The results for all profiles
of our measuring series are plotted in figure 23- All of them lie below
the curve indicated by Gruschwitz but the majority of Kehl's points also
lie below the original Gruschwitz curve. Since, however, all the results
do not greatly deviate from one another, the assumption A may be regarded
as correct in good approximation. The long dashed curve was calculated
by Pretsch (reference I5) for power profiles; the power pertaining to
a certain H-L2 1^ given in the figure at the right. Concerning the
short dashed curve, compare section 6.
Masters are different for assumption B. Gruschwitz had obtained
from his test evaluation
&P dg(6p) o -,
— — —^-^ == ari - b, with a = 8.9^+ x lO'-^ and b = h.6l x 10"^
Q dx
10 NACA TM I31U
8(^2) ^^ "^^^ total pressure at the wall distance 52:
2
6(52) = P + |[^(52)]
Because of tj = 1 - u(52) /U and the Bernoulli equation outside of the
friction layer p + Q = constant, we may also write
dg(62) dp -, d(Qii)
= — P + Q(l - il) =
dx dx"- -■ dx
Kehl, who investigated a larger Re-number range, found b to be addi-
tionally dependent on U52/v; therefore, he plots
52 dg(S2) 52 d(QTi)
b = ari ^^ = a.T\ + {h)
' Q dx ' Q dx
against U52/V • This has Deen done for our measurements in figure 2h.
In this diagram, Gruschwitz obtained a horizontal straight lihe
_->
b = constant = ii.6l X 10 and Kehl the plotted curve which, starting
o
from U52/V = 2 X 10 slowly drops. These two lines have been drawn
solidly as far as measurements existed. Our measurements show that at
least for the cases I and II - that is, for linear steep pressure rise -
b can be assumed neither as constajit nor as a unique function of U62/V
since we obtain in this diagram two essentially deviating cuj.ves. The
same result is obtained also in th^ cases III and IV for higher Re-numbers.
Only for U62/V < 10 , the deviations of the measuring points may be
interpreted as scatter. It is not the Kehl relation but the simpler
Gruschwitz relation which is confirmed here. For U52/V < 2 X 10^ Kehl
took the drawn variation of b in order to obtain significant calculation
results directly behind the transition point from laminar to turbulent
flow. The two points with U62/V < 10^ m the case V are not an argu-
ment against this b-variation for the reason that the friction layer had
been made artifically t\irbulent to start with by means of the trip wire.
Below, we shall attempt a theoretical interpretation of the relation B
which so far has been set up and investigated in a purely empirical
manner .
MCA TM 131^
11
Since the variation of the wall shearing stress proved to be very
complicated, according to figiires 2 to 6, we cannot improve upon assxjmp-
tion C which refers to the wall shearing stress. In order to estimate
the effect of this assumption on the calculation, the cases II and IV
were calculated with the aid of Walz ' s simplified integration method
under different assumptions regarding c^' . Figure 25 shows the result
for the momentum thickness 52- In the momentum theorem, we may put
^12 ~ constant since It appears only in one term 2 + H]_2 so that
even great variations of H^^g ^^^ °^ relatively small importance. In
case II, one obtains better agreement (at least up to x = 3^) between
calculation and tests by using the expression for c^ ' obtained as a
function of U52/V for the plate without pressure rise than by putting
c^' = constant. Conversely, the agreement for case IV is better with
the assumption c^' = constant. In view of the actual variation of
Cf ' which varies from case to case, no generally valid rule can there-
fore be set up for it, even before the region of the steep rise of Cf ' .
The results of the r\ calculation are represented in figure 26.
The three different calculation methods worked out by Walz are based
on the following assumptions:
Gruschwitz-Walz b = constant = k.6l X 10"3
and
c„' = constant = k x 10"3
Gruschwitz-Walz b = constant = ij-.6l X 10"^
and
Cf.' = 0.025l(U52/v)"^A
Gruschwitz-Kehl-Walz b =
0.01614-
0.85
log (U&2/V) U52/V - 300
and
= 0.0251(U52/V)"^/^
12
NACA TM I31U
As was to be expected according to figure 2U, the last, most complicated
method is precisely the one that shows the greatest deviations compared
to the experiment. In our cases II and IV, the first and simplest cal-
culation method is the best. The agreement between that calculation and
the test is still good even in the region of steep rise Cf ' . From there
on, however, large differences result.
6. ON AN ENERGY THEOREM FOR FRICTION LAYERS
Like the approximation methods for laminar boundary layers, the
Gruschwitz method is based on Karman's momentum equation which is obtained
by integration of Prandtl's boundary-layer equation with respect to the
wall distance y. One obtains thereby a statement on the total momentum
loss of the friction layer. In analogy, a statement on xhe total energy
loss in the friction layer caused by the friction can be obtained if the
boundary -layer equation
pUUjj + OVUy = -Pj^ +
Ty = PUU^ + Ty
(5)
pu.
(after addition of -p-(ux + Vy) = because of continuity) is first
multiplied by u and then integrated with respect to y
i. J%„(| „. . 1 ,2) ,,
-.8
pu^v
+ P ^ / ^x dy =
UTy dy
and because of
2
pu V
I U2v = -H U2 rS
I "Y' "- ''
J ^v ^y = / ^"^ ^^ = kf p^(l "^ - 1 ^^) ^y
(6)
This equation signifies that the loss in kinetic energy per unit length
transverse to the flow direction in the friction layer equals the rate
of doing work of the turbulent shearing stresses.
NACA TM 131^ 13
For the laminar boundary layer t = [lUy is valid so that
/ "^Uy dy = i-t / Uy2 dy here signifies the dissipation, that is, the
energy converted to heat per unit time.
In turbulent friction layers, in contrast, no simple relation between
the shearing stress and the velocity profile exists; if it did, the essen-
tially single -parameter family of tiirbulent velocity profiles woiild have
to include also a single -parameter profile class of the shearing stresses
which is not the case according to the test evaluation. If we define,
corresponding to the momentum loss thickness, an energy loss thickness
(7)
^0
and the following dimensionless
e =
work of the shearing stresses _ j _r_ S_ u , o\
work against the wall shearing stress Jq t^ Sy U
we can write equation (6) also as
e =
1
c^'u3
I^K) (9)
Since we have thus obtained one new equation with two new unknowns,
this energy theorem at first does not help us any further either. How-
ever, we can calculate the energy loss thickness 5o from the velocity
distributions and obtain, due to zhe single -parameter quality of the
velocity profiles, a fixed relation between the quantities &]_, 62, and
6^. Plotting thus the ratio H02 = B3/S2 against H-]_2 = ^±/^2 ^°''^ ^-^^
profiles measured, we obtain figure 27. All points come to lie, with
only very small deviations, on one "street." For power profiles
^/U = (y/S)"^ the long dashed curve
AHng
Ik NACA TM 1314
vould result with A = U/B and B = l/3- A good fairing curve for the
actual profiles at pressure rise and di op results if we put A = I.269
and B = 0.379' On the basis of this empirical relation, 63 may now
be calcxilated from 5]_ and 62 with sufficient accuracy.
If one more relation could be found for the ratio e as well, it
would be possible to calculate from the new equation (9) for instance
cj.' . The calculation of e from the measiiring results is made difficult
by the fact that it results as the quotient of two quajitities which can
both be fo\ind only by graphic differentiation. This explains the great
variation of the e-values plotted for the different cases in figure 28.
On the whole, e varies only comparatively little; even when c^.' for
instajice in case IV increases from 3 "to 16 X 10"^, e remaint between
0.8 and 1.2. At first glance, e seems to have, according to the
defining equation (8), the significance of an efficiency which could
not exceed 1. Actually, however, e may assume any arbitrary value,
according to the velocity and shearing stress profile concerned; in the
immediate proximity of tiirb^olent separation, above all, where Tq
vanishes, e may assume arbitrary magnitude.
Because of the inaccuracy of the calculation from the test data, it
was not possible to determine for e a relation to the other boundary-
layer quantities. However, one can set up an interesting analogy between
the energy equation and Gruschwitz ' s assumption B. If one substitutes
the above relation for H02 which was found experimentally in the energy
d52
equation and eliminates with the aid of the momentum theorem, one
dx
obtains
U„ -R dH-|p Htp(A - 2e) + 2eB
— + ^ = J^ c^' (11)
U Hi2(Hi2 - B)(Hi2 " 1) dx 2AHi2(Hi2 " 1) ^
On the other hand, one may differentiate out the Gruschwitz -Kehl relation
(assumption B), equation (k) , and v—ite
^ _!_ J^ di] dHi2 ^ b - aT]
(12)
U 2ti dH]_2 dx 2ii
regarding t) as a function of H-]^2- ^ = ^(^12)- We now assume that
the relation B follows from the energy theorem. If this is the case.
MCA TM I31U 15
the left sides of equations (ll) and (12) must, first of all, be iden-
tical. We therefore equate, as an experiment, the left sides of the
equations and obtain a differential equation for t\(^E±2) > "the solution
of which reads
P (%2 - 1)^/^^ ' ^^
"^^^"^"12 2/(1 - B) ^'3^
(H12 - B)
From the H32 - H3_2 relation, ve had found for B: B = O.379. If,
furthermore, we put, for adaptation to the test results, the number equal
to 0.986, the function H-]_2(il) is, accordirig to figure 23, quite well
satisfied by equation (l3)-
Thus, the left sides of the energy equation (ll) and of the rela-
tion B (equation (12)) seem to be identical. Then the right sides also
must be equal which - solved with respect to b - results in
Hip(A - 2e) + 2eB
b = at) + -^ Cj'tI with A = I.269
AHi2(Hio - 1)
and B = 0.379 (l^)
Therewith, a relation between qaantltiee of the velocity and of the
shearing stress profile, based on the energy theorem, has been found for
the assumption of the Gruechwitz method. In practice, of course, this
equation is of no help, either, since no data concerning the newly intro-
duced quantity e are available although one deals here, according to
the defining equation (8), with a comparatively illustrative quantity.
Thus one may conclude from equation (l4) only that it is improbable that
invariably b = constant (as Gruschwitz assumes) or that b depends
solely on U52/v (as Kehl presupposes), for b appears here as a func-
tion of the respective velocity profile [r\ and Hx2) s-s well as of e
and Cf' which likewise is not determined merely by U52/^. However,
theoretically this conclusion is not cogent, either. It is, in itself,
conceivable that the course of the t'orbulence mechanism is such that,
in spite of different previous history, the complicated relations between
T^, c^', and e have precisely the properties which cause b to be,
for every rearward position, for instance a function of U62/V only and
the test evaluation according to figure 2k actually shows that b, at
least within a certain region, hardly varies.
l6 NACA TM 131^
7. SUMMARY
The report deals with measixrements in the turbxilent friction layer
along a flat p]ate where the static pressiire from the leading edge of
the plate onward systematically rises or decreases. In case of pressure
rise, there results after a certain starting distance, a large increase
in wall shearing stress to a m\iltiple of the initial value; this has
already been briefly commented on in the report UM Nr. 6603. The further
evaluation of the test material (calculations of the shearing stresses
and mixing lengths) also gave qualitative information on this problem.
As rule of thumb, it 'can merely be said that this strong increase in
friction drag does not occur as -rr- -^^ < 2 X 10^ is valid.
^ Q dx
Furthermore, the empirical relations on which the Gruschwitz method
is based are checked with the aid of the measurements. It is again
confirmed that the tiirbulent velocity profiles form with sufficient
accuracy a single -parameter family. Gruschwitz obtains from an empirical
relation a differential equation for the variation of that parameter in
flow direction; Kehl Improved that equation by not fixing a quantity b
contained in it as constant, like Gruschwitz, but by considering it as
a function of the Re-number of the friction layer. The present measure-
ments confirm at first the simple relation b = constant. However, for
higher Re-numbers of the friction layer - in the region of strongly
increasing wall shearing stresses - different values for b resiilt,
according to the previous history of the friction layer; here a relation
of the form b = b(U62/v) is no longer sufficient, either. It is shown
that this Gruschwitz -Kehl relation can be interpreted as statement of
the energy theorem applied to friction layers. However, this energy
theorem which simply signifies that the work of the t\irbvilent shearing
stresses equals the loss of kinetic energy in the friction layer does
not provide any practical help (for instance for setting up a calculation
method) . As long as a sufficient statement for calculation of the
shearing stresses themselves is lacking, a link between the newly intro-
duced total work of the shearing stresses and the known friction layer
quantities, -r^ displacement or momentum thickness, etc., also is
lacking. For the same reason, it is not even possible to calculate the
wall shearing stressj an approximation value for it must be inserted in
the Gruschwitz method. Thus the result of the present investigation is.
NACA TM 131^ 17
on the whole, negative with respect to the problem of calciilating in
advance turbulent friction layers; however, the test material represented
in the figures might prove useful for f\irther theoretical considerations.
Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics
18 NACA TM 131^
8. REFERENCES
1. Gruschwitz, E.: Die tiu-bulente Reibungsschicht in ebener Stromung
bei Druckabfall und Lruckanstieg. Ing. -Archiv, Bd. II, Heft 3,
September 1931, PP- 321-3^6.
2. Kehl, A. : Untersuchiingen iiber konvergente und divergente, turbulente
Reibungsschichten. Ing.-Archiv, Bd. XIII, Heft 5, 19^3, pp. 293-329.
(Available as R.T.P. Translation No. 2035, British Ministry of Air-
craft Production. )
3. Walz, A. : Graphische Hilfsmittel zur Berechnung der laminaren und
turbulenten Reibungsschicht. Lilientlial-Gesellschaft fur
Luftfahrtforsch\ing Bericht S 10, 19^0, pp. h-^-lh.
h. Walz, A.: Zur theoretischen Berechnung dea Hochstauftriebsbeivertes
von Tragfluge±profilen ohne und mit Auftriebsklappen. ZWB For-
scb.ungsbericht Nr. I769, March 19^3-
5. Walz, A.: Naherungsverfahren z\ir Bei echr-'-^ng dei laminaren und
turbulenten Reibungsschicht. Untersuchungen \ind Mitteilungen
Nr. 3060.
6. Wieghardt, K.: Ueber die Wandschubspannung in turbulenten Reibungs-
schichten bei veranderlichem Aussendruck. Untersuchungen \ind
Mitteilungen Nr. 6603.
7. Wieghardt, K.: Ueber einen Energiesatz zur Berechnung laminarer
Grenzschichten. (Available from CADO, Wright -Patterson Air Force
Base, as ATI 33090.)
8. Wieghardt, K.: Staurechen und Vielfachmanometer fur Messungen in
Reibungsschichten. Technische Bericht, Bd. 11, No. 7, 19^4, p. 207.
9- Prandtl, L.: Zur turbulenten Stromung in Rohren und langs Flatten.
AVA-Ereebn. IV. Lief., 1932, p. I8.
10. Scnultz-Grunow, F.: Neues Reibungswiderstandsgesetz fur glatte
Flatten. Luftfahrtforschung, Vol. 17, No. 8, Aug. 20, 19^+0,
p. 239- (Available as NACA TM 986.)
11. Nikuradse, J.: Tvurbulente Reibiingsschichten. Publication of the
ZWB, 19^2.
12. Mangier, W.: Das Verhalten der Wandschubspannung in turbulenten
ReibuTigsschichten mit Druckajistieg. Untersuchungen und Mitteilungen
Nr. 3052, 19^3-
NACA TM 131^^- 19
13. Hartree, D. R. : On an Equation Occurring in FaUoier and Skan's
Approximate Treatment of the Equations of the Boundary Layer.
Proc. Cambridge Phil. Soc. , vol. 33, pt. 2, April 1937, PP- 223-239.
ik. Buri, A.: Eine Berechnungsgrundlage fur die turhulente Grenzschicht
bei beschleunigter und verzogerter Grundstromung. Eidgenossische
Technische Hochschule, Zurich, 1931. (Available as R.T.P,
Translation No. 2073, British Ministry of Aircraft Production.)
15. Pretsch, J. : Zur theoretischen Berechnung des Profilviderstemdes.
Jahrbuch 1938 der deutschen Luftfahrtforschung, p. I 60. (Avail-
able as NACA TM 1009 •)
20
NACA TM I31I+
0.005
0.004
0.003
0.002
0.00 f
0.006
0.004
0.003
0.002
0.001
X
U = const. = 17.8mA
16
2
3
U = const. = 33.0m/^
4 X [m] 5
16
12
4 X [m] 5
Figure 1,- Constant outer velocity, displacement, momentum, and energy
thickness; local drag coefficient.
NACA TM I31J+
21
Figure 2.- Pressure rise: Case I. Displacement, momentum, and energy
thickness; local drag coefficient.
22
NACA TM I31U
Figure 3.- Pressure rise: Case n. Displacement, momentum, and energy
thickness; local drag coefficient.
NACA TM 1314
23
Figure 4,- Pressiire rise: Case HI. Displacement, momentum, and energy
thickness; local drag coefficient.
2k
NACA ™ I31U
Figure 5.- Pressure rise; Case IV. Displacement, momentum, and energy
thickness; local drag coefficient.
NACA OM 131^
25
30
\
20
/
.+■
U
y"
^
•X.
60
p[kg/m^]
-"^
40
r
Figure 6.- Pressure drop: Case V. Displacement, momentum, and energy
thickness; local drag coefficient.
26
NACA TM I31I+
4 5 6
Q dx
Figure 7.- Cf' against — -^ for pressure rise and pressure drop;
comparison with measurements of Kehl,
NACA TM 131^^
27
w
w
0)
a,
W
W
•rH
cd
2^H
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m
o
-i-i ^
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:=)
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w
•rH
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28
NACA TM 1311+
(U
w
w
u
a
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a
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rt
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• rH
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NACA TM 131^4-
29
U-- 1 7.6 mis
• A = 0.4d7m
+ 1.067m
I.9d7m
X 4.067m
1.0
0.8
0.6
0.4
0.2
-^
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\
X
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+ 1.087 m
° 1.967m
^ 4.067m
N
V
\
V,
^^x.
-4- •-
2
10
" % '"
Figure 10.- Dimensionless shearing stress profiles for constant outer velocity.
30
NACA TM I31I+
5 S ^ E E E
t\ r\ i\ f. t\ rv.
~~: c\i C\j o^i ri "O
t~^
NACA TfL 131*+
31
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32
NACA TM I31I+
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NACA TM 131^^
33
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NACA TM 13li^
Case U
Case N
Figure 15.- Shearing stress profiles and wall tangents (Buri form parameter).
MCA TM 131^^
35
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1=1
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36
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NACA TM'131^
37
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^ nmj ^
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Figure 21.- Variation of il for all measirring series.
Figure 22.- Variation of H22 for all measuring series. (Significance of
point markings, compare fig. 23.)
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UNIVFR.'^IT-' OF n npinA ,
DOC I ,jT
IN SCIENCE UERARY
- 117011
M.vr_ovlLI.E. FL 32611-7un ubA